Inmathematics, theNewton inequalities refer to a set ofmathematical inequalities related tomathematical series. These inequalities are named afterIsaac Newton who proved the theorem in 1707.^[1] Supposea₁, a₂, ..., a_n are non-negativereal numbers and let${\displaystyle e_k}$ denote thekthelementary symmetric polynomial ina₁, a₂, ..., a_n. Then theelementary symmetric means, given by

${\displaystyle S_k=\frac{e_k}{(\genfrac{}{}{0ex}{}{n}{k})},}$

satisfy theinequality

${\displaystyle S_{k-1}{S}_{k+1}\le S_k^2.}$

Equality holdsif and only if all the numbersa_i are equal.

It can be seen thatS₁ is thearithmetic mean, andS_n is then-th power of thegeometric mean.

• Maclaurin's inequality

• Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952).Inequalities. Cambridge University Press.ISBN 978-0521358804.{{cite book}}:ISBN / Date incompatibility (help)
• D.S. BernsteinMatrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55
• Maclaurin, C. (1729)."A second letter to Martin Folks, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra".Philosophical Transactions.36 (407–416):59–96.doi:10.1098/rstl.1729.0011.
• Whiteley, J.N. (1969). "On Newton's Inequality for Real Polynomials".The American Mathematical Monthly.76 (8). The American Mathematical Monthly, Vol. 76, No. 8:905–909.doi:10.2307/2317943.JSTOR 2317943.
• Niculescu, Constantin (2000)."A New Look at Newton's Inequalities".Journal of Inequalities in Pure and Applied Mathematics.1 (2). Article 17.

• Isaac Newton
• Inequalities (mathematics)
• Symmetric functions

• CS1 errors: ISBN date